revised bcrecv6n1p47 52y2011
Available online at BCREC Website: http://bcrec.undip.ac.id
Bulletin of Chemical Reaction Engineering & Catalysis, 6 (1), 2011, 47 - 52
Research Article
Modified Mathematical Model For Neutralization System In
Stirred Tank Reactor
Ahmmed Saadi Ibrehem *
Department Chemical and Petroleum Engineering, UCSI-University, 56000 Kuala Lumpur,
MALAYSIA
Received: 1st March 2011; Revised: 28th March 2011; Accepted: 7th April 2011
Abstract
A modified model for the neutralization process of Stirred Tank Reactors (CSTR) reactor is presented in
this study. The model accounts for the effect of strong acid [HCL] flowrate and strong base [NaOH]
flowrate with the ionic concentrations of [Cl-] and [Na+] on the Ph of the system. In this work, the effect of
important reactor parameters such as ionic concentrations and acid and base flowrates on the dynamic
behavior of the CSTR is investigated and the behavior of mathematical model is compared with the
reported models for the McAvoy model and Jutila model. Moreover, the results of the model are compared
with the experimental data in terms of pH dynamic study. A good agreement is observed between our
model prediction and the actual plant data. © 2011 BCREC UNDIP. All rights reserved
Keywords: Stirred Tank Reactors, Neutralization, Mathematical model, Dynamic studies
1. Introduction
A rigorous and generally applicable method of
deriving dynamic equations for pH neutralization
in Continuous Stirred Tank Reactors (CSTRs) was
presented by McAvoy et al. ( 1972) . The research
work done by McAvoy was essential to the
development of the fundamental modelling
approach of the pH neutralization process in
CSTRs. As cited and described in other literature,
the use of the CSTR in developing the pH
neutralization model was started over 50 years ago
by Kramer (1956) and by Geerlings (1957).
However those early studies concentrated largely
on the dynamic behaviour of the pH electrode
system. Subsequently, two crucial points in
developing a pH neutralization process model
which describes the nonlinearity of the
neutralization process have emerged from
published research. The two points are as follows:-
i. Material balances in terms of hydrogen ion or
hydroxyl ion concentrations would be extremely
difficult to write down. This is due to the fact that
the dissociation of water and resultant slight
change in water concentration would have to be
accounted. ii. Instead, material balances are
performed on all other atomic species and all
additional equilibrium relationships are used. The
electroneutrality principle is used to simplify the
equations.
Jutila & Orava (1981) studied control and
estimation Algorithms for Physico-Chemical model
of pH-pocesses in Stirred Tank Reactors (CSTR).
Gustafsson (1982) and Gustafsson & Waller (1983)
studied a reinforced McAvoy’s modelling principles
for pH neutralization processes and emphasised
the fact that mass balances on the invariant
species are inherently independent of reaction
rates. As described in this paper, the “invariant
* Corresponding Author.
E-mail address: ahmadsaadi1@yahoo.com (A.S. Ibrehem)
Telp.: Tel: +60149360913
bcrec_825_2011 Copyright © 2011, BCREC, ISSN 1978-2993
Bulletin of Chemical Reaction Engineering & Catalysis, 6 (1), 2011, 48
species” is actually the species that remain
chemically unchanged by the governing of
reactions in the neutralization process whereas the
“variant species” are the species that change in the
neutralization process, such as the hydrogen ions.
Another interesting and widely used account of
work involving the modelling of a pH
neutralization process is by Wright & Kravaris
(1991). Their work provided a new approach of the
design of nonlinear controllers for pH processes by
defining an alternative equivalent control
objective. That new approach results in a control
problem that is linear. A minimal order model was
produced by assuming that the flowrate of the
titrant required to operate the reactor was
negligible in comparison with the flow rate of the
process streams.Abdul Aziz Ishak et al. (2001)
study of the dynamics and control of a semibatch
wastewater neutralization process in modeling and
simulation is presented. Zainal Ahmad & Fairuoze
Roslin (2007) provide modeling of real pH
neutralization process using multiple neural
networks (MNN) combination technique. Rosdiazli
Ibrahim (2008) provide an adequate dynamic
nonlinear pH neutralization model, based on
physical and chemical principles that can
represent the real pH neutralization plant.
Mathematical models of chemical systems were
developed for many reasons. Thus, they may be
constructed to assist in the interpretation of
experimental data, to predict the consequence of
changes of system input or operating conditions, to
deduce optimal system or operating conditions and
for control purposes. Usually there is an interest
for dynamic model made to design and/or test the
proposed control system. The dynamic and steady
state simulation model for pH neutralization
process consists of a system of equations based on
mass and charge balances on the continuous
stirred tank reactor (CSTR).
.
2. Materials and Methods
The process can be considered as a continuous
stirred tank reactor (CSTR) to neutralize a strong
acid with a strong base manipulated by a control
valve. The process consists of an influent stream
(HCl), reagent stream (NaOH) to regulate the pH
of the effluent stream, and an effluent stream. A
schematic diagram is shown in Figure (1). The data
of this reaction was taken from a previous
experimental work by Ai-Poh Loh (2006).
2.1 Assumption for the Present Model
A dynamic model of the process is obtained
from the component material balance and the
Figure 1. Schematic of a pH neutralization process
equilibrium relationship under the following
assumptions:
(1) The acid-base reactions are ionic and can be
considered to take place , with the result that
the rate of reaction can be considered. The
stirred tank process dynamics in this case
would thus be not similar to the case of mixing
or blending non reacting streams.
(2) The system is in an ideal condition without any
pollutant influence.
(3) Constant temperature of 25 0C. pH electrode
potential is temperature dependent and this
should be accounted for with high temperature
applications.
(4) Perfect mixing.
(5) No valve dynamics. Usually, the valve
dynamics are much faster than the process
dynamics and, thus, can be ignored.
(6) pH probe dynamics can be significant and are
often represented by a first order lag.
(7) The volume of the reacting mixture in the tank
is constant and equal to V.
The basic model is considered adequate for this
case study since it represents the dominant
nonlinear characteristics of this single acid-single
base continuous stirred tank reactor (CSTR)
process.
2.2 Model for Strong Acid-Strong Base System
Consider a stirred tank into which hydrochloric
acid of concentration [HCl] flows into the tank at a
rate Fa (influent stream) and is neutralized by
sodium hydroxide of concentration [NaOH] flows at
a rate Fb (reagent).
The chemical reaction of these two solutions
occurring in the stirred tank reactor is :
Copyright © 2011, BCREC, ISSN 1978-2993
Bulletin of Chemical Reaction Engineering & Catalysis, 6 (1), 2011, 49
NaOH + HCL + H2O → H2O + CL-1 + Na+ + H+ +
OHThus, the ionic concentrations of [Cl-] and [Na+]
in the outflow from the tank would be related to
the total flows Fa, Fb and to the feed
concentrations of strong acid [HCl] and strong base
[NaOH] entering the tank. Rate constant of
reaction ( k1 ) (Perry’s book, 1997).Hence, the mass
balances on NaOH and Sodium ion.
Taking the derivative of equation (9):
d[ H + ]
d [OH - ]
=
dt
dt
d[Cl - ] d [ Na + ]
dt
dt
………………………………………………………...(10)
Substituting [OH-] from the equation (7) into (10)
yields:
Cb = [NaOH]
(1)
d[ H + ]
Kw d[H + ]
+
=
[ H + ]2 dt
dt
d[Cl - ] d [ Na + ]
dt
dt
………………………………………………………..(11)
(2)
Mass balance on HCl and Chloride ion
Ca = [HCl]
d [ H + ] æ [ H + ]2 + K w ö
ç
÷ =
dt è [ H + ]2 ø
………………………………………………………..(12)
(3)
Substituting of equations (2) and (4) into equation
(12) gives:
+ 2
(4)
Acids and bases have free hydrogen and
hydroxyl ions, respectively. Since the relationship
between hydrogen ions and hydroxide ions in a
given solution is constant for a given set of
conditions, either one can be determined by
knowing the other.
[ H + ] [OH - ] = K w = 10-14
[O H - ] =
(5)
Kw
[H + ]
……………………………………………………...(6)
Taking the derivative of equation (6):
K d[ H + ]
d [OH - ] d æ K w ö
= ç + ÷ = - +w 2
dt
dt è [ H ] ø
[H ]
dt
……………………………………………………..(7)
Electroneutrality balance
[H+ ] + [Na+ ] = [Cl- ]+ [OH- ]
……………………………………………………..(8)
Rearranging equation (8) as:
+
+
[ H ] - [OH ] = [Cl ] - [ Na ]
-
-
d[Cl - ] d [ Na + ]
dt
dt
………………………………………………………….(9)
=
+ 2
1
[ {−
+
−
1
{−
+
(
−
(
+
+
)} +
)} +
1
1
]
………………………………………………………...(13)
We can rearrange last equation as follows;
…………………………………………………(14)
After rearranging;
Substituting [OH-] from the equation (8) into (14)
yields:
………………………………………………………...(15)
Equation (15) can be solved numerically for
[H+], while the pH is a logarithmic function; a
change of one pH unit represents a ten-fold change
in hydrogen ion concentration:
pH = - log [H+]
(16)
3. Model solution and analysis
The previously described process model
equations (1-16) incorporating the parameter
values of Table (1) were solved in Matlab using the
Copyright © 2011, BCREC, ISSN 1978-2993
Bulletin of Chemical Reaction Engineering & Catalysis, 6 (1), 2011, 50
Table 1. Estimation of the reactor model parameters for mathematical model system
8
Fb
Fb
Fb
Fb
7
Value and Unit
0.000001 mol/liter
0.000001 mol/liter
0.016667 liter/s
0.016667 liter/s
10-14
30 liter
0.001 sec-1
25 oC
1atm
6
pH
Symbol
Ca
Cb
Fa
Fb
Kw
V
k1
Temperature
Pressure
Fa
Fa
Fa
Fa
6
5
4
3
2
0
2000
4000
6000
8000
Time (sec)
Figure 3. Effects of base flowrate in the pH with
variable time
8
7
= 0.018 liter/sec
= 0.02 liter/sec
= 0.015 liter/sec
= 0.013 liter/sec
= 0.018 liter/sec
=0.02 liter/sec
=0.015 liter/sec
= 0.013 liter/sec
9
8
pH
pH
7
5
6
5
4
Ca
Ca
Ca
Ca
4
3
=0.000011mol/liter
= 0.000012 mol/liter
= 0.000009 mol/liter
= 0.000008 mol/liter
3
2
2
0
2000
4000
6000
8000
0
2000
4000
6000
8000
10000
Time (sec)
Time (sec)
Figure 2. Effects of acid flowrate in the pH with
variable time
Differential Algebraic Equation solver (DAE) with
the fourth order Runge-kutta method. Physical
constants and operating parameters for the
mathematical model system were used in Table (1).
The process was simulated for the effects of
[NaOH], [HCl], [Na+] and [Cl-] on pH with variable
time.
In the following sections the simulation results
are described for the different concentrations of the
system. Figure (2) shows the effect of [HCl] flow
rate on the pH of the system. The pH profile has
an inverse relationship with the increase in flow
rate of [HCl]. Figure (3) shows the effect of [NaOH]
flow rate on the pH of the system. The pH profile
has an proporational relationship with the increase
in flow rate of [NaOH].
Figures (4) and (5) show the effect of
concentrations [Cl-] and [Na+] on the pH with
respect to time. It can be seen that the pH depends
on the concentrations [Cl-] and [Na+]. The pH has
Figure 4. Effects of concentation of [Cl-] in the pH
with variable time
an inverse relationship with the increase in
concentrations [Cl-] as shown in Figure (4). The pH
has proporational relationship with the increase in
concentrations [Na+] as shown in Figure (5). All
these behavior are mention above can be
represented in Figures (4) and (5) respectively.
4. Model validation with previous models and
experimental data
Comparison of the two previously available
models; McAvoy model and Jutila model and
modified model in terms of their dynamic
predications is shown in Figures (6),(7),(8) and (9).
The figures indicates that the predications of the
three models are close to each other at the startup
conditions of the reactor operation. System works
at nominal condition as in Table (1) then at time
3000 sec increases acid flowrate to 10% and makes
comparison between experimental results of
number 1 versus model predicted pH and two
Copyright © 2011, BCREC, ISSN 1978-2993
Bulletin of Chemical Reaction Engineering & Catalysis, 6 (1), 2011, 51
8.5
9
8.0
8
7.5
7
7.0
pH
pH
6
6.5
Experiment results
Modified model
McAvoy model
Jutila model
6.0
5
Cb = 0.000011 mol/liter
Cb = 0.000012 mol/liter
Cb = 0.000009 mol/liter
Cb = 0.000008 mol/liter
4
5.5
5.0
3
4.5
0
2
0
2000
4000
6000
8000
2000
4000
Time (sec)
Figure 5. Effects of concentation of [Na+] in the
pH with variable time
6000
8000
10000
Time (sec)
10000
Figure 7. Experimental results versus model
predicted pH using -10% a step change of the
acid flow rate in the system
9.5
9.0
7.0
8.5
pH
6.5
pH
6.0
8.0
Experiment results
McAvoy model
Jutila model
Modified model
7.5
5.5
experiment results
McAvoy model
Jutila model
Modified model
5.0
4.5
7.0
6.5
0
2000
4000
6000
8000
10000
Time (sec)
4.0
0
2000
4000
6000
8000
Time (sec)
Figure 6. Experimental results versus model
predicted pH using +10% a step change of the
acid flow rate in the system
10000
Figure 8. Experimental results versus model
predicted pH using +10% a step change of the
base flow rate in the system
7.5
7.0
pH
6.5
6.0
Experiment results
Modified model
Jutila model
McAvoy model
5.5
5.0
0
2000
4000
6000
8000
Time (sec)
Figure 9. Experimental results versus model
predicted pH using -10% a step change of the
base flow rate in the system
Copyright © 2011, BCREC, ISSN 1978-2993
10000
Bulletin of Chemical Reaction Engineering & Catalysis, 6 (1), 2011, 52
previous models as shown in Figure (6).
Experiment number 2 the system works at
nominal conditions then at time 3000 s decreases
acid flowrate to 10% and makes comparison
between experimental results versus model
predicted pH and two previous models as shown in
Figure (7).
In experiment 3 the system works at nominal
conditions then at time 3000 s increases base
flowrate to 10% and makes comparison between
experimental results versus model predicted pH
and two previous models as shown in Figure (8). In
experiment 4 the system works at nominal
conditions then at time 3000 s decreases base
flowrate to 10% and makes comparison between
experimental results versus model predicted pH
and two previous models as shown in Figure (9).
However this behavior changes as the process
dynamics proceeds in time and by the end of the
time of reaction, the modified model becomes closer
to Jutila model and McAvoy model. The close
behavior of modified mathematical model to the
Jutila model is mainly due to the fact that the
same consideration for the rate reaction about
concentrations of ions happens in the process.
In summary the dynamic behavior of the
modified model is very close to Jutila model and
McAvoy model in the initial stages and starts to
differ with change in the time. The accuracy of the
dynamic behavior of these models can be seen from
their comparison with actual experimental data
[Loh et al., 2001] as shown in Figures (6),(7),(8)
and (9).
present model which is very close to that of the
McAvoy model and Jutila model in the initial
stages but little different with change in time. The
model developed here will also be used in modelbased prediction control to control the reactor
which is part of our future work.
References
[1]
Ai-Poh Loh,, Dhruba Sankar De, and P. R.
Krishnaswamy,
5. Conclusion
A modified dynamic structure model was
developed in this work. This model takes into
account the presence of acid and bases in the
reaction with ions which depend on chemical
reactions of acid and bases concentrations feeds. In
addition, the concentrations effect of acid and
bases on the system were included. Model
simulations indicate that it is capable of
predicating reactor performance indicators as well
as calculating the changes of ions through chemical
of the reaction. The model presented in this work
was compared with two previously available
models and results of the proposed model were
compared with experimental data of neutralization
process. From its observed accuracy, we can
conveniently use this model as a predictive tool to
study the effects of operating, kinetic and
hydrodynamics parameters on the reactor
performance. The comparison results between the
modified model and the other two available models
gave good indication about the behavior of the
Copyright © 2011, BCREC, ISSN 1978-2993
Bulletin of Chemical Reaction Engineering & Catalysis, 6 (1), 2011, 47 - 52
Research Article
Modified Mathematical Model For Neutralization System In
Stirred Tank Reactor
Ahmmed Saadi Ibrehem *
Department Chemical and Petroleum Engineering, UCSI-University, 56000 Kuala Lumpur,
MALAYSIA
Received: 1st March 2011; Revised: 28th March 2011; Accepted: 7th April 2011
Abstract
A modified model for the neutralization process of Stirred Tank Reactors (CSTR) reactor is presented in
this study. The model accounts for the effect of strong acid [HCL] flowrate and strong base [NaOH]
flowrate with the ionic concentrations of [Cl-] and [Na+] on the Ph of the system. In this work, the effect of
important reactor parameters such as ionic concentrations and acid and base flowrates on the dynamic
behavior of the CSTR is investigated and the behavior of mathematical model is compared with the
reported models for the McAvoy model and Jutila model. Moreover, the results of the model are compared
with the experimental data in terms of pH dynamic study. A good agreement is observed between our
model prediction and the actual plant data. © 2011 BCREC UNDIP. All rights reserved
Keywords: Stirred Tank Reactors, Neutralization, Mathematical model, Dynamic studies
1. Introduction
A rigorous and generally applicable method of
deriving dynamic equations for pH neutralization
in Continuous Stirred Tank Reactors (CSTRs) was
presented by McAvoy et al. ( 1972) . The research
work done by McAvoy was essential to the
development of the fundamental modelling
approach of the pH neutralization process in
CSTRs. As cited and described in other literature,
the use of the CSTR in developing the pH
neutralization model was started over 50 years ago
by Kramer (1956) and by Geerlings (1957).
However those early studies concentrated largely
on the dynamic behaviour of the pH electrode
system. Subsequently, two crucial points in
developing a pH neutralization process model
which describes the nonlinearity of the
neutralization process have emerged from
published research. The two points are as follows:-
i. Material balances in terms of hydrogen ion or
hydroxyl ion concentrations would be extremely
difficult to write down. This is due to the fact that
the dissociation of water and resultant slight
change in water concentration would have to be
accounted. ii. Instead, material balances are
performed on all other atomic species and all
additional equilibrium relationships are used. The
electroneutrality principle is used to simplify the
equations.
Jutila & Orava (1981) studied control and
estimation Algorithms for Physico-Chemical model
of pH-pocesses in Stirred Tank Reactors (CSTR).
Gustafsson (1982) and Gustafsson & Waller (1983)
studied a reinforced McAvoy’s modelling principles
for pH neutralization processes and emphasised
the fact that mass balances on the invariant
species are inherently independent of reaction
rates. As described in this paper, the “invariant
* Corresponding Author.
E-mail address: ahmadsaadi1@yahoo.com (A.S. Ibrehem)
Telp.: Tel: +60149360913
bcrec_825_2011 Copyright © 2011, BCREC, ISSN 1978-2993
Bulletin of Chemical Reaction Engineering & Catalysis, 6 (1), 2011, 48
species” is actually the species that remain
chemically unchanged by the governing of
reactions in the neutralization process whereas the
“variant species” are the species that change in the
neutralization process, such as the hydrogen ions.
Another interesting and widely used account of
work involving the modelling of a pH
neutralization process is by Wright & Kravaris
(1991). Their work provided a new approach of the
design of nonlinear controllers for pH processes by
defining an alternative equivalent control
objective. That new approach results in a control
problem that is linear. A minimal order model was
produced by assuming that the flowrate of the
titrant required to operate the reactor was
negligible in comparison with the flow rate of the
process streams.Abdul Aziz Ishak et al. (2001)
study of the dynamics and control of a semibatch
wastewater neutralization process in modeling and
simulation is presented. Zainal Ahmad & Fairuoze
Roslin (2007) provide modeling of real pH
neutralization process using multiple neural
networks (MNN) combination technique. Rosdiazli
Ibrahim (2008) provide an adequate dynamic
nonlinear pH neutralization model, based on
physical and chemical principles that can
represent the real pH neutralization plant.
Mathematical models of chemical systems were
developed for many reasons. Thus, they may be
constructed to assist in the interpretation of
experimental data, to predict the consequence of
changes of system input or operating conditions, to
deduce optimal system or operating conditions and
for control purposes. Usually there is an interest
for dynamic model made to design and/or test the
proposed control system. The dynamic and steady
state simulation model for pH neutralization
process consists of a system of equations based on
mass and charge balances on the continuous
stirred tank reactor (CSTR).
.
2. Materials and Methods
The process can be considered as a continuous
stirred tank reactor (CSTR) to neutralize a strong
acid with a strong base manipulated by a control
valve. The process consists of an influent stream
(HCl), reagent stream (NaOH) to regulate the pH
of the effluent stream, and an effluent stream. A
schematic diagram is shown in Figure (1). The data
of this reaction was taken from a previous
experimental work by Ai-Poh Loh (2006).
2.1 Assumption for the Present Model
A dynamic model of the process is obtained
from the component material balance and the
Figure 1. Schematic of a pH neutralization process
equilibrium relationship under the following
assumptions:
(1) The acid-base reactions are ionic and can be
considered to take place , with the result that
the rate of reaction can be considered. The
stirred tank process dynamics in this case
would thus be not similar to the case of mixing
or blending non reacting streams.
(2) The system is in an ideal condition without any
pollutant influence.
(3) Constant temperature of 25 0C. pH electrode
potential is temperature dependent and this
should be accounted for with high temperature
applications.
(4) Perfect mixing.
(5) No valve dynamics. Usually, the valve
dynamics are much faster than the process
dynamics and, thus, can be ignored.
(6) pH probe dynamics can be significant and are
often represented by a first order lag.
(7) The volume of the reacting mixture in the tank
is constant and equal to V.
The basic model is considered adequate for this
case study since it represents the dominant
nonlinear characteristics of this single acid-single
base continuous stirred tank reactor (CSTR)
process.
2.2 Model for Strong Acid-Strong Base System
Consider a stirred tank into which hydrochloric
acid of concentration [HCl] flows into the tank at a
rate Fa (influent stream) and is neutralized by
sodium hydroxide of concentration [NaOH] flows at
a rate Fb (reagent).
The chemical reaction of these two solutions
occurring in the stirred tank reactor is :
Copyright © 2011, BCREC, ISSN 1978-2993
Bulletin of Chemical Reaction Engineering & Catalysis, 6 (1), 2011, 49
NaOH + HCL + H2O → H2O + CL-1 + Na+ + H+ +
OHThus, the ionic concentrations of [Cl-] and [Na+]
in the outflow from the tank would be related to
the total flows Fa, Fb and to the feed
concentrations of strong acid [HCl] and strong base
[NaOH] entering the tank. Rate constant of
reaction ( k1 ) (Perry’s book, 1997).Hence, the mass
balances on NaOH and Sodium ion.
Taking the derivative of equation (9):
d[ H + ]
d [OH - ]
=
dt
dt
d[Cl - ] d [ Na + ]
dt
dt
………………………………………………………...(10)
Substituting [OH-] from the equation (7) into (10)
yields:
Cb = [NaOH]
(1)
d[ H + ]
Kw d[H + ]
+
=
[ H + ]2 dt
dt
d[Cl - ] d [ Na + ]
dt
dt
………………………………………………………..(11)
(2)
Mass balance on HCl and Chloride ion
Ca = [HCl]
d [ H + ] æ [ H + ]2 + K w ö
ç
÷ =
dt è [ H + ]2 ø
………………………………………………………..(12)
(3)
Substituting of equations (2) and (4) into equation
(12) gives:
+ 2
(4)
Acids and bases have free hydrogen and
hydroxyl ions, respectively. Since the relationship
between hydrogen ions and hydroxide ions in a
given solution is constant for a given set of
conditions, either one can be determined by
knowing the other.
[ H + ] [OH - ] = K w = 10-14
[O H - ] =
(5)
Kw
[H + ]
……………………………………………………...(6)
Taking the derivative of equation (6):
K d[ H + ]
d [OH - ] d æ K w ö
= ç + ÷ = - +w 2
dt
dt è [ H ] ø
[H ]
dt
……………………………………………………..(7)
Electroneutrality balance
[H+ ] + [Na+ ] = [Cl- ]+ [OH- ]
……………………………………………………..(8)
Rearranging equation (8) as:
+
+
[ H ] - [OH ] = [Cl ] - [ Na ]
-
-
d[Cl - ] d [ Na + ]
dt
dt
………………………………………………………….(9)
=
+ 2
1
[ {−
+
−
1
{−
+
(
−
(
+
+
)} +
)} +
1
1
]
………………………………………………………...(13)
We can rearrange last equation as follows;
…………………………………………………(14)
After rearranging;
Substituting [OH-] from the equation (8) into (14)
yields:
………………………………………………………...(15)
Equation (15) can be solved numerically for
[H+], while the pH is a logarithmic function; a
change of one pH unit represents a ten-fold change
in hydrogen ion concentration:
pH = - log [H+]
(16)
3. Model solution and analysis
The previously described process model
equations (1-16) incorporating the parameter
values of Table (1) were solved in Matlab using the
Copyright © 2011, BCREC, ISSN 1978-2993
Bulletin of Chemical Reaction Engineering & Catalysis, 6 (1), 2011, 50
Table 1. Estimation of the reactor model parameters for mathematical model system
8
Fb
Fb
Fb
Fb
7
Value and Unit
0.000001 mol/liter
0.000001 mol/liter
0.016667 liter/s
0.016667 liter/s
10-14
30 liter
0.001 sec-1
25 oC
1atm
6
pH
Symbol
Ca
Cb
Fa
Fb
Kw
V
k1
Temperature
Pressure
Fa
Fa
Fa
Fa
6
5
4
3
2
0
2000
4000
6000
8000
Time (sec)
Figure 3. Effects of base flowrate in the pH with
variable time
8
7
= 0.018 liter/sec
= 0.02 liter/sec
= 0.015 liter/sec
= 0.013 liter/sec
= 0.018 liter/sec
=0.02 liter/sec
=0.015 liter/sec
= 0.013 liter/sec
9
8
pH
pH
7
5
6
5
4
Ca
Ca
Ca
Ca
4
3
=0.000011mol/liter
= 0.000012 mol/liter
= 0.000009 mol/liter
= 0.000008 mol/liter
3
2
2
0
2000
4000
6000
8000
0
2000
4000
6000
8000
10000
Time (sec)
Time (sec)
Figure 2. Effects of acid flowrate in the pH with
variable time
Differential Algebraic Equation solver (DAE) with
the fourth order Runge-kutta method. Physical
constants and operating parameters for the
mathematical model system were used in Table (1).
The process was simulated for the effects of
[NaOH], [HCl], [Na+] and [Cl-] on pH with variable
time.
In the following sections the simulation results
are described for the different concentrations of the
system. Figure (2) shows the effect of [HCl] flow
rate on the pH of the system. The pH profile has
an inverse relationship with the increase in flow
rate of [HCl]. Figure (3) shows the effect of [NaOH]
flow rate on the pH of the system. The pH profile
has an proporational relationship with the increase
in flow rate of [NaOH].
Figures (4) and (5) show the effect of
concentrations [Cl-] and [Na+] on the pH with
respect to time. It can be seen that the pH depends
on the concentrations [Cl-] and [Na+]. The pH has
Figure 4. Effects of concentation of [Cl-] in the pH
with variable time
an inverse relationship with the increase in
concentrations [Cl-] as shown in Figure (4). The pH
has proporational relationship with the increase in
concentrations [Na+] as shown in Figure (5). All
these behavior are mention above can be
represented in Figures (4) and (5) respectively.
4. Model validation with previous models and
experimental data
Comparison of the two previously available
models; McAvoy model and Jutila model and
modified model in terms of their dynamic
predications is shown in Figures (6),(7),(8) and (9).
The figures indicates that the predications of the
three models are close to each other at the startup
conditions of the reactor operation. System works
at nominal condition as in Table (1) then at time
3000 sec increases acid flowrate to 10% and makes
comparison between experimental results of
number 1 versus model predicted pH and two
Copyright © 2011, BCREC, ISSN 1978-2993
Bulletin of Chemical Reaction Engineering & Catalysis, 6 (1), 2011, 51
8.5
9
8.0
8
7.5
7
7.0
pH
pH
6
6.5
Experiment results
Modified model
McAvoy model
Jutila model
6.0
5
Cb = 0.000011 mol/liter
Cb = 0.000012 mol/liter
Cb = 0.000009 mol/liter
Cb = 0.000008 mol/liter
4
5.5
5.0
3
4.5
0
2
0
2000
4000
6000
8000
2000
4000
Time (sec)
Figure 5. Effects of concentation of [Na+] in the
pH with variable time
6000
8000
10000
Time (sec)
10000
Figure 7. Experimental results versus model
predicted pH using -10% a step change of the
acid flow rate in the system
9.5
9.0
7.0
8.5
pH
6.5
pH
6.0
8.0
Experiment results
McAvoy model
Jutila model
Modified model
7.5
5.5
experiment results
McAvoy model
Jutila model
Modified model
5.0
4.5
7.0
6.5
0
2000
4000
6000
8000
10000
Time (sec)
4.0
0
2000
4000
6000
8000
Time (sec)
Figure 6. Experimental results versus model
predicted pH using +10% a step change of the
acid flow rate in the system
10000
Figure 8. Experimental results versus model
predicted pH using +10% a step change of the
base flow rate in the system
7.5
7.0
pH
6.5
6.0
Experiment results
Modified model
Jutila model
McAvoy model
5.5
5.0
0
2000
4000
6000
8000
Time (sec)
Figure 9. Experimental results versus model
predicted pH using -10% a step change of the
base flow rate in the system
Copyright © 2011, BCREC, ISSN 1978-2993
10000
Bulletin of Chemical Reaction Engineering & Catalysis, 6 (1), 2011, 52
previous models as shown in Figure (6).
Experiment number 2 the system works at
nominal conditions then at time 3000 s decreases
acid flowrate to 10% and makes comparison
between experimental results versus model
predicted pH and two previous models as shown in
Figure (7).
In experiment 3 the system works at nominal
conditions then at time 3000 s increases base
flowrate to 10% and makes comparison between
experimental results versus model predicted pH
and two previous models as shown in Figure (8). In
experiment 4 the system works at nominal
conditions then at time 3000 s decreases base
flowrate to 10% and makes comparison between
experimental results versus model predicted pH
and two previous models as shown in Figure (9).
However this behavior changes as the process
dynamics proceeds in time and by the end of the
time of reaction, the modified model becomes closer
to Jutila model and McAvoy model. The close
behavior of modified mathematical model to the
Jutila model is mainly due to the fact that the
same consideration for the rate reaction about
concentrations of ions happens in the process.
In summary the dynamic behavior of the
modified model is very close to Jutila model and
McAvoy model in the initial stages and starts to
differ with change in the time. The accuracy of the
dynamic behavior of these models can be seen from
their comparison with actual experimental data
[Loh et al., 2001] as shown in Figures (6),(7),(8)
and (9).
present model which is very close to that of the
McAvoy model and Jutila model in the initial
stages but little different with change in time. The
model developed here will also be used in modelbased prediction control to control the reactor
which is part of our future work.
References
[1]
Ai-Poh Loh,, Dhruba Sankar De, and P. R.
Krishnaswamy,
5. Conclusion
A modified dynamic structure model was
developed in this work. This model takes into
account the presence of acid and bases in the
reaction with ions which depend on chemical
reactions of acid and bases concentrations feeds. In
addition, the concentrations effect of acid and
bases on the system were included. Model
simulations indicate that it is capable of
predicating reactor performance indicators as well
as calculating the changes of ions through chemical
of the reaction. The model presented in this work
was compared with two previously available
models and results of the proposed model were
compared with experimental data of neutralization
process. From its observed accuracy, we can
conveniently use this model as a predictive tool to
study the effects of operating, kinetic and
hydrodynamics parameters on the reactor
performance. The comparison results between the
modified model and the other two available models
gave good indication about the behavior of the
Copyright © 2011, BCREC, ISSN 1978-2993